| L(s) = 1 | + (−51.0 − 62.9i)3-s − 459.·7-s + (−1.35e3 + 6.41e3i)9-s + 9.13e3i·11-s + 4.80e4·13-s + 5.33e4i·17-s − 1.22e5·19-s + (2.34e4 + 2.88e4i)21-s + 2.63e5i·23-s + (4.72e5 − 2.42e5i)27-s − 1.04e6i·29-s + 1.39e6·31-s + (5.74e5 − 4.66e5i)33-s − 2.84e6·37-s + (−2.45e6 − 3.02e6i)39-s + ⋯ |
| L(s) = 1 | + (−0.629 − 0.776i)3-s − 0.191·7-s + (−0.206 + 0.978i)9-s + 0.624i·11-s + 1.68·13-s + 0.638i·17-s − 0.938·19-s + (0.120 + 0.148i)21-s + 0.943i·23-s + (0.889 − 0.456i)27-s − 1.47i·29-s + 1.51·31-s + (0.484 − 0.393i)33-s − 1.52·37-s + (−1.06 − 1.30i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.776i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.629 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.4558323939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4558323939\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (51.0 + 62.9i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 459.T + 5.76e6T^{2} \) |
| 11 | \( 1 - 9.13e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 4.80e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 5.33e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.22e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 2.63e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 1.04e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.39e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + 2.84e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 2.57e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 1.51e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 2.59e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 2.74e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.78e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 8.28e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 4.62e5T + 4.06e14T^{2} \) |
| 71 | \( 1 + 3.15e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.02e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 2.02e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 3.52e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 4.31e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 5.78e7T + 7.83e15T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.82381491451153277468912616624, −9.909830092823357781531745339973, −8.540638809429402443914451378068, −7.85233118072984441563387122203, −6.53461492463379259859107542517, −6.12535704772911396894483436733, −4.82661655165382176247934895229, −3.60318897361462799046255504105, −2.06669979703095363975280156488, −1.17085066500228300414566231229,
0.11507188354642929694376748798, 1.21418133745373966854087963937, 3.02021913067330475725495484118, 3.94015103846849846122088907833, 5.01812901384892721111673321610, 6.08904676844071354205700361603, 6.74952914211700751542149247223, 8.485256249497555068818533642553, 8.948180100040157940138311231783, 10.29948928550106066122801621528
